Abstract

The finite sum of the squares of the Mie coefficients is very useful for addressing problems of classical light scattering. An approximate formula available in the literature, and still in use today, has been developed to determine a priori the number of the most significant terms needed to evaluate the scattering cross section. Here, we obtain an improved formula, which includes the number of terms needed for determining the scattering cross section within a prescribed relative error. This is accomplished using extended precision computation for a wide range of commonly used size parameters and indices of refraction. The revised formula for the finite number of terms can be a promising and valuable approach for efficient modeling light scattering phenomena.

Log-log plot evidencing the functional dependence of the maximum number of terms (N) as a function of size factor (x) for m=2. Each curve is plotted for different truncation errors from 10−5 to 10−155, from bottom to top, respectively. Dashed and dot-dash blue lines represent trends, and the dashed black line represents Wiscombe’s criterion Eq. (5).

Contour plot of the number of terms (N) as a function of size factor (x) and accuracy (ε) for m=2. First contour (left) corresponds to (N=50), and each successive contour adds another 50 terms to the series. Horizontal lines represent maximum accuracy for finite precision computation.